Question

after an injection, the amount of a medication a, in cubic - centimeters (cc), in the bloodstream decreases with time t in hours. suppose that under certain conditions a is given by the function below, where a₀ is the initial amount of the medication. assume that an initial amount of 40 cc is injected. complete parts (a) through (d)
a(t)=\frac{a₀}{t² + 1}
a) find a(0), a(1), a(2), a(7), and a(10)
a(0)=
(type an integer or decimal rounded to the nearest ten - thousandth as needed.)

Explanation:

Step1: Substitute \(t = 0\) into the function

Given \(A(t)=\frac{A_0}{t^{2}+1}\) and \(A_0 = 40\), when \(t = 0\), we have \(A(0)=\frac{40}{0^{2}+1}\).

$$A(0)=\frac{40}{1}=40$$

Step2: Substitute \(t = 1\) into the function

When \(t = 1\), \(A(1)=\frac{40}{1^{2}+1}\).

$$A(1)=\frac{40}{2}=20$$

Step3: Substitute \(t = 2\) into the function

When \(t = 2\), \(A(2)=\frac{40}{2^{2}+1}\).

$$A(2)=\frac{40}{4 + 1}=\frac{40}{5}=8$$

Step4: Substitute \(t = 7\) into the function

When \(t = 7\), \(A(7)=\frac{40}{7^{2}+1}\).

$$A(7)=\frac{40}{49+1}=\frac{40}{50}=0.8$$

Step5: Substitute \(t = 10\) into the function

When \(t = 10\), \(A(10)=\frac{40}{10^{2}+1}\).

$$A(10)=\frac{40}{100 + 1}=\frac{40}{101}\approx0.3960$$

Answer:

\(A(0)=40\), \(A(1)=20\), \(A(2)=8\), \(A(7)=0.8\), \(A(10)\approx0.3960\)